Briefly explained, Saul's authority did not come from the full Sanhedrin, but from the High Priest, himself, as head of the Jews's religion (Judaism). The Sanhedrin would be involved after Saul brought any prisoners back to Jerusalem. The High Priest's authority in Jewish religious matters extended beyond Roman administrative boundaries and was not affected by the regular limitations of Roman civil law. More details are given below:
[Act 9:1-2 KJV] "1 And Saul, yet breathing out threatenings and slaughter against the disciples of the Lord, went unto the high priest, 2 And desired of him letters to Damascus to the synagogues, that if he found any of this way, whether they were men or women, he might bring them bound unto Jerusalem."
The early followers of Christ had yet to bear the title "Christians," which came later (Acts 11:26). At the time of Saul's journey to Damascus they were referred to as "the way" or "this way" (Acts 9:2). Following the stoning of Stephen, they were treated as an aberrant and dangerous sect of Judaism (Acts 8:1-4) but not a separate religion.
Saul’s letters of authority to arrest any followers of "this way" in the Damascene synagogues and bring them back to Jerusalem for trial and punishment were granted within the sphere of Judaism, which was outside of regular Roman laws or courts and extended beyond the province of Judea into synagogues anywhere in the empire.
The High Priest exercised his authority under the general Roman policy of toleration toward the Jews. Judaism had been tolerated in Rome by diplomatic treaty with Graeco-Judaean (Hasmonean) rulers during the later days of the Roman Republic (161 B.C.) when Judea sought protection and aid in its struggle against the Seleucid rulers (I Maccabees 8:17-20; and Josephus, Antiquities, 13. 9:2). Rome’s toleration continued in the days of Julius Caesar “because their ancestral laws predated Rome. Jews had legal privileges as a collegia (defined by Roman law as religious & legal entities), giving them the right to assemble, have common meals and property, govern and tax themselves, and enforce their own discipline.” (http://www.yashanet.com/studies/romstudy/rom1.htm) Toleration by Rome toward Jews was reiterated in the Edict of Augustus in 1 B.C., which protected practice of their “own customs in accordance with their ancestral law” in the Temple and the synagogues ( Edict of Augustus, Josephus, Antiquities 16.162–5). In particular, there was a very Jew-tolerant attitude by the Romans in the latter years of Tiberias (the setting of Acts Chapter 9) in reaction to the fall of Sejanus, the Jew-hating Praetorian Prefect: "Therefore, all people in every country, even if they were not naturally well inclined towards the Jewish nation, took great care not to violate or attack any of the Jewish customs of laws" (Philo, De Legatione ad Gaium, xxiv).
My Personal Speculation: Of course, Saul would encounter the risen Lord Jesus Christ on the road to Damascus, so it never came to be actually seen what the Roman civil authorities might have done in reaction to the High Priest extending his authority in this affair. Given that he did give Saul letters to be used in the arrest of early followers of Christ, and Saul acted upon that authority, it appears to me, at least, that he was acting within the scope of his office, perhaps in an "act first, answer questions later," move if anyone had appealed to the Romans. A hint of what the Roman reaction might have been is seen years later, when Saul, now called Paul, was accused of violating the Temple, a religious offense, and placed before the Sanhedrin (Acts Chapters 22 and 23). Later, after Paul had been brought before the Roman governor, the charges changed. The first charges brought against him were political and civil, that of sedition, because Paul's accusers knew the Romans were not primarily interested in the Jewish religious laws. (Acts 24:5-8) (They did something similar in the trial of Jesus before Pilate. When Pilate failed to be moved by their claims that Jesus had violated religious law, the Jewish leaders changed tactics, pressuring Pilate by stating that Jesus claimed to be a king, which could be treason against the emperor, a political charge a Roman governor could not ignore.) Paul languished in Roman custody as a political pawn (Acts 24:27) and because the governor sought a bribe (Acts 24:26). Paul's only recourse was to appeal his case to Caesar, a right he had as a Roman citizen (Acts 25:10 and 11).